Augmenting a pure and hybrid vertical equilibrium scheme via data-driven surrogate modelling

TL;DR

Vertical equilibrium models offer speed but limited applicability to equilibrium regions; coupling with full-dimensional models adds overhead. This work employs fast surrogates based on linear regression and spline interpolation to accelerate the gas plume distance $z_p$, coarse-level mobilities, and secondary variables within both pure VE and hybrid VE-FD schemes, while preserving mass conservation. Across three test setups, the approach achieves speedups from $18\%$ to $75.4\%$ with negligible solution error, and the combined surrogate deployment can even outperform a full FD simulation in runtime. The study provides open-source code and discusses boundaries for surrogate reuse and potential extensions to improve VE-based subsurface simulations for storage and energy applications.

Abstract

Vertical equilibrium (VE) models have been introduced as computationally efficient alternatives to traditional mass and momentum balance equations for fluid flow in porous media. Since VE models are only accurate in regions where phase equilibrium holds, while traditional simulations are computationally demanding, hybrid methods have been proposed to combine the accuracy of the full-dimensional approach with the efficiency of VE model. However, coupling both models introduces computational overhead that can make hybrid simulations slower than fully traditional ones. To address this, we introduce data-driven surrogates to predict the gas plume distance and coarse-level mobilities in the VE model, as well as predictors to accelerate the coupling scheme. We focus on surrogate models with short inference times to minimize computational overhead during frequent function calls. The proposed approach preserves key physical properties, such as mass conservation, while substantially reducing simulation runtimes. Overall, combining data-driven methods with the hybrid VE scheme yields an enhanced model that outperforms traditional simulations in speed while introducing only negligible errors.

Figures (7)

• Figure 1: Illustration of the gas plume distance $z_p$ in a 2D slice for a gas plume with caprocks at the top and bottom
• Figure 2: Coupling concept between the FD (yellow) and the VE (red) domain across a coupling interface (blue line) with domain height $H=z_T-z_B$. The coupling fluxes are computed on the fine level and if necessary accumulated to a coarse-level flux entering the VE subdomain.
• Figure 3: Boundary conditions, geometries and solution fields for three test cases. The first column depicts boundary conditions and impermeable lenses, while the second column illustrates the saturation fields at the end of the simulations. For the hybrid models (\ref{['fig_case_b', 'fig_case_c']}), the FD subdomain is denoted via element edges in the solution field, while the smooth surface represents the VE subdomain
• Figure 4: Target relative gas plume distance and proposed saturation-based features. The relative plume distance is defined as the gas plume distance normalized by the domain height $H=z_T-z_B$
• Figure 5: Comparison of target coarse-level mobilities with Brooks–Corey mobilities computed using fixed $\lambda=2$